The challenge is to have significant populations of Punic and Vandalic speakers in North Africa that are large enough to later form independent states, yet dwarfed compared to speakers of Amazigh and African Latin (including descendent languages).
The first POD entails thousands of Carthaginians managed to survive being killed or enslaved in the aftermath of the Third Punic War to remain a large presence in the area.
The second POD entails the Vandal Kingdom managing to escape conquest by the Byzantines long enough for their language to take hold over the population they ruled over, along with a No/Screwed Islam scenario.